A fast singly diagonally implicit runge–kutta method for solving 1D unsteady convection‐diffusion equations

Abstract

In this article, a fast singly diagonally implicit Runge–Kutta method is designed to solve unsteady one‐dimensional convection diffusion equations. We use a three point compact finite difference approximation for the spatial discretization and also a three‐stage singly diagonally implicit Runge–Kutta (RK) method for the temporal discretization. In particular, a formulation evaluating the boundary values assigned to the internal stages for the RK method is derived so that a phenomenon of the order of the reduction for the convergence does not occur. The proposed scheme not only has fourth‐order accuracy in both space and time variables but also is computationally efficient, requiring only a linear matrix solver for a tridiagonal matrix system. It is also shown that the proposed scheme is unconditionally stable and suitable for stiff problems. Several numerical examples are solved by the new scheme and the numerical efficiency and superiority of it are compared with the numerical results obtained by other methods in the literature. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 788–812, 2014

High order locally one-dimensional methods for solving two-dimensional parabolic equations

“…ρ(H 1 ) and ρ(H 2 ) are the spectral radii of H 1 and H 2 , respectively. This shows that the new developed schemes (22) and (33) are unconditionally stable bypassing the accuracy barrier theorem [27].…”

“…22, we may complete the entire calculation from t k to t k+1 . Due to the use of fourth order scheme for discretizing the space variables and (2, 2) Padé approximation for the temporal variable, it is not difficult to find that equations in (22) are of fourth order accuracy in both time and space.…”

“…All these schemes obtain fourth order accuracy in space, but only second order accuracy in time, since the Crank-Nicolson implicit method is employed for time discretization. Very recently, a semi-discrete method and Padé approximations or the Runge-Kutta methods were exploited to increase the temporal accuracy [19][20][21][22][23][24]. Vu and Alexander [19] developed a series of explicit exponential Runge-Kutta methods of high order for parabolic problems.…”

Flux Transport Characteristics of Free Boundary Value Problems for a Class of Generalized Convection-Diffusion Equation

“…It is very easy to obtain the value of by applying (1) + (1) = 0 in (18)- (20). Based on (18)- (20) and the corresponding we can easily describe the graph of the Considering 0 < 1 < 2 , 0 < < 1, we have 1 > 2 . Figure 4 indicates that the diffusion flux decreases with the decrease of ( ).…”

“…On the one hand, some scholars consider the existence, uniqueness, or nonuniqueness of solutions for the convection-diffusion equations (for example, see [5][6][7][8][9][10][11][12][13]). On the other hand, others focus on the numerical solution of the convection-diffusion equation by all kinds of methods, for instance, the spectral element method [14], the finite element method [15][16][17], the finite difference method [18,19], and the Runge-Kutta method [20].…”